According to the proposed OAM-SM scheme, the selections of the transmission antenna, the OAM state, and the radiated symbol are based on the random input bits. The input bits are assumed to follow independent uniform distributions. Therefore, each transmission antenna is selected with the same probability \({1 \mathord {\left\bad.{\vphantom{1 M}} \right.\hspace{0.0pt}} M}\) . Similarly, each OAM state is selected with a probability \({1 \mathord {\left\bad.{\vphantom{1 L}} \right.\hspace{0.0pt}} L}\)  and each radiated symbol is selected with a probability \({1 \mathord {\left\bad.{\vphantom{1 P}} \right.\hspace{0.0pt}} P}\) , respectively [1]}. Thus, the PDF of the signal received by the first antenna of the \(m^{\prime } - th\)  receive antenna pair, i.e., \({y_{m^{\prime },1}}\) , is derived by
\({\begin{array}{c}\mathcal {P}\left( {{y_{m^{\prime },1}}} \right) = \sum \limits _{m = 1}^M {\sum \limits _{\ell = 1}^L {\sum \limits _{p = 1}^P {\mathcal {P}\left( {{y_{m^{\prime },1}},m,l,{x_p}} \right)} } } \hfill \\\;\;\;\;\;\;\;\;\;\;\;\;\;\; = \sum \limits _{m = 1}^M {\sum \limits _{\ell = 1}^L {\sum \limits _{p = 1}^P {\mathcal {P}\left( {{y_{m^{\prime },1}}\left| {m,l,{x_p}} \right.} \right)} } } \mathcal {P}\left( m \right)\mathcal {P}\left( l \right)\mathcal {P}\left( {{x_p}} \right) \hfill \\\;\;\;\;\;\;\;\;\;\;\;\;\;\; = \frac{1}{{MLP}}\sum \limits _{m = 1}^M {\sum \limits _{\ell = 1}^L {\sum \limits _{p = 1}^P {\frac{1}{{\pi \sigma _{\mathbf {w}}^2}}\exp \left( { - \frac{{{{\left| {{y_{m^{\prime },1}} - \sqrt{\rho }h_{mm^{\prime },1}^l{x_p}} \right|}^2}}}{{\sigma _{\mathbf {w}}^2}}} \right)} } } \hfill \\\end{array}} .\qquad \mathrm {(21)}\) 
